def test_coefficient_vector(self):
mesh = (np.arange(3), np.arange(10, 12))
G = Grid.Grid(mesh, initializer=lambda *x: 0.0)
# print G
F = FD.FiniteDifferenceEngineADI(G,
coefficients={
(0, 1): lambda t, *x: 1
})
op = self.F.operators[(0, 1)]
op.D.data[op.D.data != 0] = op.D.data[op.D.data != 0]**0
op.D = todia(op.D)
# diamat = op.D
# densemat = diamat.todense()
# f0 = lambda t, *x: x[0]
# f1 = lambda t, *x: x[1]
f = lambda t, *x: x
vec = F.coefficient_vector(f, 0, 0)
ref = np.tile(mesh[0], G.shape[1]), np.repeat(mesh[1], G.shape[0])
npt.assert_array_equal(ref, vec, "Failed when focused on dim: %i", 0)
vec = F.coefficient_vector(f, 0, 1)
ref = np.repeat(mesh[0], G.shape[1]), np.tile(mesh[1], G.shape[0])
npt.assert_array_equal(ref, vec, "Failed when focused on dim: %i", 1)
def setUp(self):
# print "Setting up Params for CPP tests"
shape = (5,4)
self.v1 = np.arange(shape[0]*shape[1], dtype=float)**2
self.v2 = self.v1.copy()
self.v2.resize(shape)
coeffs = {(0,) : lambda t, *x: x[0]*x[1],
(0,0): lambda t, *x: x[0]*x[1],
(1,) : lambda t, *x: x[0]*x[1],
(1,1): lambda t, *x: x[0]*x[1],
(0,1): lambda t, *x: x[0]*x[1],
}
bounds = {
(0,) : ((0, lambda *args: 0), (1, lambda *args: 1)),
(0,0) : ((0, lambda *args: 0), (None, lambda *args: 1)),
(1,) : ((None, lambda *args: None), (None, lambda *args: None)),
(1,1) : ((1, lambda *args: 0.0), (None, lambda *args: None)),
}
schemes = {}
self.G = Grid.Grid([np.arange(shape[0], dtype=float)**3, np.arange(shape[1], dtype=float)**3], lambda x, y: (x*shape[1]+y)**2)
self.F = FD.FiniteDifferenceEngineADI(self.G, coefficients=coeffs,
boundaries=bounds, schemes=schemes, force_bandwidth=None)
# print "Setting up FDE for CPP tests"
self.F.init()
self.F.operators[0].R = np.arange(self.G.size, dtype=float)
self.F.operators[1].R = np.arange(self.G.size, dtype=float)
def build_FDE_2D(bounds=bounds):
def init(x, y):
mesh = np.sqrt(x[:, np.newaxis]**2 + y**2)
return heat_kernel_nD(2, 0, mesh, dt)
G = Grid(mesh=(np.linspace(-1, 1, 21), np.linspace(-1, 1, 21)),
initializer=init)
# vis.surface(G.domain[-1], *G.mesh)
F = FD.FiniteDifferenceEngineADI(G, coefficients=coeffs, boundaries=bounds)
return F
def build_FDE_1D(bounds=bounds):
def init(x):
return x**2
# mesh = x
# return heat_kernel_nD(2, 0, mesh, start)
G = Grid(mesh=(np.linspace(xmin, xmax, nx), ), initializer=init)
# vis.lineplot(G.domain[-1], *G.mesh)
F = FD.FiniteDifferenceEngineADI(G, coefficients=coeffs, boundaries=bounds)
return F
def setUp(self):
s1_enable = 1
s2_enable = 1
v1_enable = 1
v2_enable = 1
r_enable = 1
dirichlet_s = 1
dirichlet_v = 1
kappa = 1
r = 0.06
theta = 0.04
sigma = 0.4
rho = 0.6
spot_max = 1500.0
var_max = 13.0
nspots = 5
nvols = 5
spotdensity = 7.0 # infinity is linear?
varexp = 5
up_or_down_spot = 'up'
up_or_down_var = 'down'
flip_idx_spot = nspots // 2
flip_idx_var = nvols // 2
k = spot_max / 4.0
# spots = np.linspace(0, spot_max, nspots)
# vars = np.linspace(0, var_max, nvols)
spots = utils.sinh_space(k, spot_max, spotdensity, nspots)
vars = utils.exponential_space(0.00, 0.04, var_max, varexp, nvols)
def mu_s(t, *dim):
return r * dim[0] * s1_enable
def gamma2_s(t, *dim):
return 0.5 * dim[1] * dim[0]**2 * s2_enable
def mu_v(t, *dim):
return kappa * (theta - dim[1]) * v1_enable
def gamma2_v(t, *dim):
return 0.5 * sigma**2 * dim[1] * v2_enable
coeffs = {
(): lambda t: -r * r_enable,
(0, ): mu_s,
(0, 0): gamma2_s,
(1, ): mu_v,
(1, 1): gamma2_v,
(0, 1): lambda t, *dim: dim[0] * dim[1] * rho * sigma
}
coeffs = {k: lambda *x: 1 for k in coeffs}
bounds = {
(0, ): ((0, lambda *args: 0), (1, lambda *args: 1)),
(0, 0): ((0, lambda *args: 0), (None, lambda *args: 1)),
(1, ): ((None, lambda *args: None), (None, lambda *args: None)),
(1, 1): ((1, lambda *args: 0.0), (None, lambda *args: None)),
}
schemes = {}
# G = Grid.Grid((spots, vars), initializer=lambda x0,x1: np.maximum(x0-k,0))
self.G = Grid.Grid((spots, vars), initializer=lambda x0, x1: x0 * x1)
# print G
self.F = FD.FiniteDifferenceEngineADI(self.G,
coefficients=coeffs,
boundaries=bounds,
schemes=schemes,
force_bandwidth=None)
self.F.init()
# self.F.operators[1].diagonalize()
self.FG = FDG.FiniteDifferenceEngineADI()
self.FG.from_host_FiniteDifferenceEngine(self.F)
def test_dirichlet_boundary(self):
spots = np.arange(5.)
vars = np.arange(4.)
def rndgrid(x0, x1):
return np.random.random((x0.shape[0], x1.shape[1]))
G = Grid.Grid((spots, vars), initializer=rndgrid)
npt.assert_array_equal(G.shape, tuple(map(len, G.mesh)))
coeffs = bounds = {}
coeffs = {
(0, ): lambda t, *dim: 1,
# (0,0): lambda t, *dim: 0,
(
1, ): lambda t, *dim: 0,
# (1,1): lambda t, *dim: 0,
# (0,1): lambda t, *dim: 0
}
bounds = {
(0, ): ((0, lambda t, *x: 1), (0, lambda t, *x: 1)),
# (0,0): ((None, lambda *x: 1), (0, lambda *x: 1)),
(
1, ): ((0, lambda *x: 1), (0, lambda *x: 1)),
# (1,1): ((0, lambda *x: 1), (0, lambda *x: 1)),
}
F = FD.FiniteDifferenceEngineADI(G,
coefficients=coeffs,
boundaries=bounds,
force_bandwidth=None)
F.init()
for d, o in F.simple_operators.items():
l = [[1] * F.grid.shape[(o.axis + 1) % 2]] * 2
npt.assert_array_equal(
o.dirichlet,
l,
err_msg="Dim: %s, dirichlet: %s, expected: %s" %
(d, o.dirichlet, l))
for d, o in F.simple_operators.items():
l = [[1] * F.grid.shape[(o.axis + 1) % 2]] * 2
npt.assert_array_equal(
o.dirichlet,
l,
err_msg="Dim: %s, dirichlet: %s, expected: %s" %
(d, o.dirichlet, l))
for d in bounds.keys():
B = F.simple_operators[d]
# print "Derivative", d, "Dirichlets", B.dirichlet
g = (B + 1).solve(F.grid.domain[-1])
if B.axis == 1:
g = g.T
if B.dirichlet[0] is not None:
npt.assert_array_equal(g[0, :], 1)
if B.dirichlet[1] is not None:
npt.assert_array_equal(g[-1, :], 1)
for d in bounds.keys():
B = F.simple_operators[d]
# print "Derivative", d, "Dirichlets", B.dirichlet
# print B.dirichlet
g = B.apply(F.grid.domain[-1])
if B.axis == 1:
g = g.T
if B.dirichlet[0] is not None:
npt.assert_array_equal(g[0, :], 1)
if B.dirichlet[1] is not None:
npt.assert_array_equal(g[-1, :], 1)
for d in bounds.keys():
B = F.simple_operators[d]
# print "Derivative", d, "Dirichlets", B.dirichlet
# fp(B.data)
g = (B + 1).solve(F.grid.domain[-1])
# fp(g)
if B.axis == 1:
g = g.T
if B.dirichlet[0] is not None:
npt.assert_array_equal(g[0, :], 1)
if B.dirichlet[1] is not None:
npt.assert_array_equal(g[-1, :], 1)
def test_numpy_vs_operators(self):
spots = np.linspace(0, 1, 4)
vars = np.linspace(0, 10, 4)
# spots = self.G.mesh[0]
# vars = self.G.mesh[1]
G = Grid.Grid((spots, vars), initializer=lambda x0, x1: x1 * 1)
coeffs = {
(): lambda t: 0,
(0, ): lambda t, *dim: dim[0],
(0, 0): lambda t, *dim: dim[0],
(1, ): lambda t, *dim: 0 * dim[1],
(1, 1): lambda t, *dim: dim[1],
(0, 1): lambda t, *dim: dim[0] * dim[1]
}
bounds = {
(0, ): ((None, lambda *x: None), (None, lambda *x: 3)),
(0, 0): ((None, lambda *x: None), (None, lambda *x: 3)),
(1, ): ((None, lambda *x: None), (None, lambda *x: 1)),
(1, 1): ((None, lambda *x: None), (None, lambda *x: 1)),
}
F = FD.FiniteDifferenceEngineADI(G,
coefficients=coeffs,
boundaries=bounds,
force_bandwidth=None)
F.init()
cb = utils.clear_boundary
g = F.grid.domain[-1]
op_ = {'delta': {}, 'grid_delta': {}, 'derivative': {}}
np_ = {'delta': {}, 'grid_delta': {}, 'derivative': {}}
dims = coeffs.keys()
dims.remove(())
for dim in dims:
op_['grid_delta'][dim] = F.simple_operators[dim].copy()
op_['grid_delta'][dim].R = None
op_['delta'][(0, )] = op_['grid_delta'][(0, )].deltas[:, np.newaxis]
op_['delta'][(1, )] = op_['grid_delta'][(1, )].deltas
x = F.grid.mesh[0]
y = F.grid.mesh[1]
np_['delta'][(0, )] = np.gradient(x)[:, np.newaxis]
np_['delta'][(1, )] = np.gradient(y)
np_['delta'][(0, )][0, 0] = np.nan
np_['delta'][(1, )][0] = np.nan
for f in (op_, np_):
f['delta'][(0, 0)] = f['delta'][(0, )]**2
f['delta'][(1, 1)] = f['delta'][(1, )]**2
f['delta'][(0, 1)] = f['delta'][(1, )] * f['delta'][(0, )]
f['delta'][(1, 0)] = f['delta'][(1, )] * f['delta'][(0, )]
np_['grid_delta'][(0, )], np_['grid_delta'][(1, )] = np.gradient(g)
for fst in [0, 1]:
np_['grid_delta'][(fst,
0)], np_['grid_delta'][(fst, 1)] = np.gradient(
np_['grid_delta'][(fst, )])
Y, X = np.meshgrid(y, x)
for dim in dims:
op_['derivative'][dim] = cb(op_['grid_delta'][dim].apply(g),
inplace=True)
npt.assert_(op_['derivative'][dim].shape == g.shape)
np_['derivative'][dim] = cb(np_['grid_delta'][dim] /
np_['delta'][dim],
inplace=True)
np_['derivative'][dim] *= F.coefficients[dim](0, X, Y)
op_['derivative'][dim] *= F.coefficients[dim](0, X, Y)
np_['derivative'][(1, 0)] = cb(np_['grid_delta'][(1, 0)] /
np_['delta'][(1, 0)],
inplace=True)
np_['derivative'][(1, 0)] *= F.coefficients[(0, 1)](0, X, Y)
for dim in dims:
for x in ('delta', 'derivative'):
msg = "Comparing %s in dimension %s" % (x, dim)
try:
npt.assert_array_almost_equal(op_[x][dim],
np_[x][dim],
decimal=7,
err_msg=msg)
except AssertionError:
print msg, op_['grid_delta'][dim].axis
fp((op_['grid_delta'][dim]).D.data)
fp((op_[x][dim]), fmt='f')
fp((np_[x][dim]), fmt='f')
fp((op_[x][dim] - np_[x][dim]), fmt='f')
# npt.assert_allclose(op_[x][dim], np_[x][dim], err_msg=msg)
npt.assert_array_almost_equal(op_[x][dim],
np_[x][dim],
decimal=7,
err_msg=msg)
npt.assert_array_almost_equal(np_['derivative'][(0, 1)],
np_['derivative'][(1, 0)],
err_msg=msg)
def setUp(self):
s1_enable = 1
s2_enable = 1
v1_enable = 1
v2_enable = 1
r_enable = 1
dirichlet_s = 1
dirichlet_v = 1
kappa = 1
r = 0.06
theta = 0.04
sigma = 0.4
rho = 0.6
spot_max = 1500.0
var_max = 13.0
nspots = 5
nvols = 5
spotdensity = 7.0 # infinity is linear?
varexp = 5
# TODO:!!!!XXX TODO XXX
# var_max = nvols-1
# spot_max = nspots-1
up_or_down_spot = 'up'
up_or_down_var = 'down'
flip_idx_spot = nspots // 2
flip_idx_var = nvols // 2
k = spot_max / 4.0
# spots = np.linspace(0, spot_max, nspots)
# vars = np.linspace(0, var_max, nvols)
spots = utils.sinh_space(k, spot_max, spotdensity, nspots)
vars = utils.exponential_space(0.00, 0.04, var_max, varexp, nvols)
def mu_s(t, *dim):
return r * dim[0] * s1_enable
def gamma2_s(t, *dim):
return 0.5 * dim[1] * dim[0]**2 * s2_enable
def mu_v(t, *dim):
return kappa * (theta - dim[1]) * v1_enable
def gamma2_v(t, *dim):
return 0.5 * sigma**2 * dim[1] * v2_enable
coeffs = {
(): lambda t: -r * r_enable,
(0, ): mu_s,
(0, 0): gamma2_s,
(1, ): mu_v,
(1, 1): gamma2_v,
(0, 1): lambda t, *dim: dim[0] * dim[1] * rho * sigma
}
bounds = {
# D: U = 0 VN: dU/dS = 1
(
0, ): ((0 if dirichlet_s else 1, lambda *args: 0.0),
(1, lambda *args: 1.0)),
# D: U = 0 Free boundary
(0, 0): ((0 if dirichlet_s else 1, lambda *args: 0.0),
(None, lambda *x: 1.0)),
# Free boundary at low variance
(
1, ): (
(None, lambda *x: None),
# (0.0, lambda t, *dim: 0),
# D intrinsic value at high variance
# (None, lambda *x: None)
(0 if dirichlet_v else 1,
lambda t, *dim: np.maximum(0.0, dim[0] - k))),
# # Free boundary
(1, 1): (
(1, lambda *x: 0),
# D intrinsic value at high variance
# (None, lambda *x: None)
(0 if dirichlet_v else 1,
lambda t, *dim: np.maximum(0.0, dim[0] - k)))
}
schemes = {}
schemes[(0, )] = ({
"scheme": "center"
}, {
"scheme": up_or_down_spot,
"from": flip_idx_spot
})
schemes[(1, )] = ({
"scheme": "center"
}, {
"scheme": up_or_down_var,
"from": flip_idx_var
})
dss = np.hstack((np.nan, np.diff(spots)))
dvs = np.hstack((np.nan, np.diff(vars)))
As_ = utils.nonuniform_complete_coefficients(
dss, up_or_down=up_or_down_spot, flip_idx=flip_idx_spot)[0]
Ass_ = utils.nonuniform_complete_coefficients(dss)[1]
L1_ = []
R1_ = []
# As_, Ass_ = utils.nonuniform_forward_coefficients(dss)
assert (not np.isnan(As_.data).any())
assert (not np.isnan(Ass_.data).any())
for j, v in enumerate(vars):
# Be careful not to overwrite operators
As, Ass = As_.copy(), Ass_.copy()
m = 2
mu_s = r * spots * s1_enable
gamma2_s = 0.5 * v * spots**2 * s2_enable
for i, z in enumerate(mu_s):
# print z, coeffs[0,](0, spots[i])
assert z == coeffs[0, ](0, spots[i])
for i, z in enumerate(gamma2_s):
# print z, coeffs[0,0](0, spots[i], v)
assert z == coeffs[0, 0](0, spots[i], v)
Rs = np.zeros(nspots)
Rs[-1] = 1
As.data[m - 2, 2:] *= mu_s[:-2]
As.data[m - 1, 1:] *= mu_s[:-1]
As.data[m, :] *= mu_s
As.data[m + 1, :-1] *= mu_s[1:]
As.data[m + 2, :-2] *= mu_s[2:]
Rs *= mu_s
Rss = np.zeros(nspots)
Rss[-1] = 2 * dss[-1] / dss[-1]**2
Ass.data[m + 1, -2] = 2 / dss[-1]**2
Ass.data[m, -1] = -2 / dss[-1]**2
Ass.data[m - 2, 2:] *= gamma2_s[:-2]
Ass.data[m - 1, 1:] *= gamma2_s[:-1]
Ass.data[m, :] *= gamma2_s
Ass.data[m + 1, :-1] *= gamma2_s[1:]
Ass.data[m + 2, :-2] *= gamma2_s[2:]
Rss *= gamma2_s
L1_.append(As.copy())
L1_[j].data += Ass.data
L1_[j].data[m, :] -= 0.5 * r * r_enable
L1_[j].data[m, 0] = 1 * dirichlet_s
R1_.append((Rs + Rss).copy())
R1_[j][0] = 0
L2_ = []
R2_ = []
Av_ = utils.nonuniform_complete_coefficients(dvs,
up_or_down=up_or_down_var,
flip_idx=flip_idx_var)[0]
Avv_ = utils.nonuniform_complete_coefficients(dvs)[1]
assert (not np.isnan(Av_.data).any())
assert (not np.isnan(Avv_.data).any())
for i, s in enumerate(spots):
mu_v = kappa * (theta - vars) * v1_enable
gamma2_v = 0.5 * sigma**2 * vars * v2_enable
# for j, z in enumerate(mu_v):
# assert z == coeffs[1,](0, 0, vars[j])
# for j, z in enumerate(gamma2_v):
# assert z == coeffs[1,1](0, 0, vars[j])
# Be careful not to overwrite our operators
Av, Avv = Av_.copy(), Avv_.copy()
m = 2
Av.data[m - 2, 2] = -dvs[1] / (dvs[2] * (dvs[1] + dvs[2]))
Av.data[m - 1, 1] = (dvs[1] + dvs[2]) / (dvs[1] * dvs[2])
Av.data[m,
0] = (-2 * dvs[1] - dvs[2]) / (dvs[1] * (dvs[1] + dvs[2]))
Av.data[m - 2, 2:] *= mu_v[:-2]
Av.data[m - 1, 1:] *= mu_v[:-1]
Av.data[m, :] *= mu_v
Av.data[m + 1, :-1] *= mu_v[1:]
Av.data[m + 2, :-2] *= mu_v[2:]
Rv = np.zeros(nvols)
Rv *= mu_v
Avv.data[m - 1, 1] = 2 / dvs[1]**2
Avv.data[m, 0] = -2 / dvs[1]**2
Avv.data[m - 2, 2:] *= gamma2_v[:-2]
Avv.data[m - 1, 1:] *= gamma2_v[:-1]
Avv.data[m, :] *= gamma2_v
Avv.data[m + 1, :-1] *= gamma2_v[1:]
Avv.data[m + 2, :-2] *= gamma2_v[2:]
Rvv = np.zeros(nvols)
Rvv[0] = 2 * dvs[1] / dvs[1]**2
Rvv *= gamma2_v
L2_.append(Av.copy())
L2_[i].data += Avv.data
L2_[i].data[m, :] -= 0.5 * r * r_enable
L2_[i].data[
m,
-1] = 1 * dirichlet_v # This is to cancel out the previous value so we can
# set the dirichlet boundary condition using R.
# Then we have U_i + -U_i + R
R2_.append(Rv + Rvv)
R2_[i][-1] = 0 # np.maximum(0, s - k)
def flatten_tensor(mats):
diags = np.hstack([x.data for x in mats])
flatmat = scipy.sparse.dia_matrix(
(diags, mats[0].offsets),
shape=(diags.shape[1], diags.shape[1]))
return flatmat
L1 = flatten_tensor(L1_)
L2 = flatten_tensor(L2_)
R1 = np.array(R1_).T
R2 = np.array(R2_)
self.As_ = As_
self.Ass_ = Ass_
self.L1_ = L1
self.R1_ = R1
self.L2_ = L2
self.R2_ = R2
# G = Grid.Grid((spots, vars), initializer=lambda x0,x1: np.maximum(x0-k,0))
self.G = Grid.Grid((spots, vars), initializer=lambda x0, x1: x0 * x1)
# print G
self.F = FD.FiniteDifferenceEngineADI(self.G,
coefficients=coeffs,
boundaries=bounds,
schemes=schemes,
force_bandwidth=None)
self.F.init()